In Exercises 59–94, solve each absolute value inequality. |x| > 3

Verified step by step guidance

1. Understand the absolute value inequality \(|x| > 3\). This means that the distance of \(x\) from 0 on the number line is greater than 3.
2. Rewrite the inequality as two separate inequalities: \(x > 3\) or \(x < -3\).
3. These inequalities represent the solution set where \(x\) is either greater than 3 or less than -3.
4. Graphically, this means that the solution is all values of \(x\) to the right of 3 and to the left of -3 on the number line.
5. Express the solution in interval notation: \((-\infty, -3) \cup (3, \infty)\).

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is denoted as |x|. This means |x| is equal to x if x is positive or zero, and |x| is equal to -x if x is negative. Understanding absolute value is crucial for solving inequalities that involve it.

Inequalities

Inequalities express a relationship between two values that are not necessarily equal. They can be represented using symbols such as >, <, ≥, and ≤. In the context of absolute value inequalities, the solution set often includes multiple intervals, which must be determined by analyzing the conditions set by the inequality. This requires a solid grasp of how to manipulate and interpret inequalities.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the number line. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the solution to the inequality |x| > 3 can be expressed in interval notation as (-∞, -3) ∪ (3, ∞), indicating all values less than -3 and greater than 3.